Transport in Porous Media

, Volume 119, Issue 3, pp 499–519 | Cite as

A Heuristic Insight on End-Point Calculation and a New Phase Interference Parameter in Two-Phase Relative Permeability Curves for Horizontal Fracture Flow

  • Mohammad Ranjbaran
  • Saeed Shad
  • Vahid Taghikhani
  • Shahab Ayatollahi


Relative permeability curves of two-phase flow in a fracture have been a subject of study in recent years. The importance of these curves have been widely observed in multidisciplines, such as water subsurface resources, geothermal energy and underground hydrocarbon resources, especially fractured oil and gas reservoirs. Extensive experimental studies have been cited alongside the numerical studies in this area. However, simple analytical and practical solutions are still attractive. In the current study, wettability effects and phase interference explicitly were tried to be implemented in a simple analytical formula. The wettability effects are represented by residual saturations which resulted in direct calculation of relative permeability end points. In addition, the phase interference part affected the shape of the curves that allowed to quantify the degree of phase interference from no phase interference, assigned as zero, to ultimate phase interference, assigned as infinity. The results were compared and validated with the available experimental data in the literature. The proposed formulation is applicable for both smooth and rough fracture assemblies.


Analytical study Phase interference Smooth and rough fractures Two-phase flow 


  1. Alturki, A., Maini, B., Gates, I.: The effect of wall roughness on two-phase flow in a rough-walled Hele-Shaw cell. J. Pet. Explor. Prod. Technol. 4(4), 397–426 (2014). doi: 10.1007/s13202-013-0090-x CrossRefGoogle Scholar
  2. Babadagli, T., Raza, S., Ren, X., Develi, K.: Effect of surface roughness and lithology on the water-gas and water-oil relative permeability ratios of oil-wet single fractures. Int. J. Multiph. Flow 75, 68–81 (2015a). doi: 10.1016/j.ijmultiphaseflow.2015.05.005
  3. Babadagli, T., Ren, X., Develi, K.: Effects of fractal surface roughness and lithology on single and multiphase flow in a single fracture: an experimental investigation. Int. J. Multiph. Flow 68, 40–58 (2015b). doi: 10.1016/j.ijmultiphaseflow.2014.10.004 CrossRefGoogle Scholar
  4. Bertels, S.P., DiCarlo, D.A., Blunt, M.J.: Measurement of aperture distribution, capillary pressure, relative permeability, and in situ saturation in a rock fracture using computed tomography scanning. Water Resour. Res. 37(3), 649–662 (2001). doi: 10.1029/2000WR900316 CrossRefGoogle Scholar
  5. Bird, R.B., Stewart, W.E., Lightfoot, E.N.: Transport Phenomena, 2nd edn. Wiley, New York (2002)Google Scholar
  6. Brown, S.R.: Fluid flow through rock joints: the effect of surface roughness. J. Geophys. Res. Solid Earth 92(B2), 1337–1347 (1987). doi: 10.1029/JB092iB02p01337 CrossRefGoogle Scholar
  7. Chen, C.Y., Horne, R.N.: Two-phase flow in rough-walled fractures: experiments and a flow structure model. Water Resour. Res. 42(3), W03–430 (2006). doi: 10.1029/2004WR003837 Google Scholar
  8. Chen, C.Y., Horne, R.N., Fourar, M.: Experimental study of liquid–gas flow structure effects on relative permeabilities in a fracture. Water Resour. Res. 40(8), W08–301 (2004). doi: 10.1029/2004WR003026 CrossRefGoogle Scholar
  9. Durlofsky, L., Brady, J.F.: Analysis of the Brinkman equation as a model for flow in porous media. Phys. Fluids 30(11), 3329–3334 (1987). doi: 10.1063/1.866465 CrossRefGoogle Scholar
  10. Fourar, M., Lenormand, R.: A viscous coupling model for relative permeabilities in fractures. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (1998). doi: 10.2118/49006-MS
  11. Gross, S., Reusken, A.: Numerical Methods for Two-Phase Incompressible Flows, vol. 40. Springer, Berlin (2011)Google Scholar
  12. Hanks, R.W.: The laminar-turbulent transition for flow in pipes, concentric annuli, and parallel plates. AIChE J. 9(1), 45–48 (1963). doi: 10.1002/aic.690090110 CrossRefGoogle Scholar
  13. Honarpour, M.M., Koederitz, F., Herbert, A.: Relative Permeability of Petroleum Reservoirs. CRC Press Inc, Boca Raton (1986)Google Scholar
  14. Huo, D., Benson, S.M.: Experimental investigation of stress-dependency of relative permeability in rock fractures. Transp. Porous Med. 113(3), 567–590 (2016). doi: 10.1007/s11242-016-0713-z CrossRefGoogle Scholar
  15. Lian, P., Cheng, L., Ma, C.Y.: The characteristics of relative permeability curves in naturally fractured carbonate reservoirs. J. Can. Pet. Technol. 51(02), 137–142 (2012). doi: 10.2118/154814-PA CrossRefGoogle Scholar
  16. Liu, H.H., Wei, M.Y., Rutqvist, J.: Normal-stress dependence of fracture hydraulic properties including two-phase flow properties. Hydrogeol. J. 21(2), 371–382 (2013). doi: 10.1007/s10040-012-0915-6 CrossRefGoogle Scholar
  17. Lomize, G.: Flow in fractured rocks. Gosenergoizdat Mosc. 127, 197 (1951)Google Scholar
  18. Pan, X.: Immiscible two-phase flow in a fracture. Ph.D. thesis, University of Calgary, Canada (1999)Google Scholar
  19. Pan, X., Wong, R., Maini, B., et al.: Steady state immiscible oil and water flow in a smooth-walled fracture. J. Can. Pet. Technol. 37(05), 52–59 (1998). doi: 10.2118/98-05-04 CrossRefGoogle Scholar
  20. Patir, N., Cheng, H.: An average flow model for determining effects of three-dimensional roughness on partial hydrodynamic lubrication. J. lubr. Technol. 100(1), 12–17 (1978). doi: 10.1115/1.3453103 CrossRefGoogle Scholar
  21. Persoff, P., Pruess, K.: Two-phase flow visualization and relative permeability measurement in natural rough-walled rock fractures. Water Resour. Res. 31(5), 1175–1186 (1995). doi: 10.1029/95WR00171 CrossRefGoogle Scholar
  22. Pruess, K., Tsang, Y.: On two-phase relative permeability and capillary pressure of rough-walled rock fractures. Water Resour. Res. 26(9), 1915–1926 (1990). doi: 10.1029/WR026i009p01915 CrossRefGoogle Scholar
  23. Pyrak-Nolte, L.J., Cook, N.G., Nolte, D.D.: Fluid percolation through single fractures. Geophys. Res. Lett. 15(11), 1247–1250 (1988). doi: 10.1029/GL015i011p01247 CrossRefGoogle Scholar
  24. Rangel-German, E., Akin, S., Castanier, L.: Multiphase-flow properties of fractured porous media. J. Pet. Sci. Eng. 51(3), 197–213 (2006). doi: 10.1016/j.petrol.2005.12.010 CrossRefGoogle Scholar
  25. Raza, S., Hejazi, S.H., Gates, I.D.: Two phase flow of liquids in a narrow gap: phase interference and hysteresis. Phys. Fluids 28(7), 074–102 (2016). doi: 10.1063/1.4953238 CrossRefGoogle Scholar
  26. Renshaw, C.E.: On the relationship between mechanical and hydraulic apertures in rough-walled fractures. J. Geophys. Res. Solid Earth 100(B12), 24629–24636 (1995). doi: 10.1029/95JB02159 CrossRefGoogle Scholar
  27. Romm, E.: Flow Characteristics of Fractured Rocks. Nedra, Moscow (1966)Google Scholar
  28. Saboorian-Jooybari, H.: Analytical estimation of water-oil relative permeabilities through fractures. Oil Gas Sci. Technol. Rev. dIFP Energ. Nouv. 71(3), 31 (2016). doi: 10.2516/ogst/2014054 CrossRefGoogle Scholar
  29. Saltelli, A., Chan, K., Scott, E.M., et al.: Sensitivity Analysis, vol. 1. Wiley, New York (2000)Google Scholar
  30. Shad, S., Gates, I.D.: Multiphase flow in fractures: co-current and counter-current flow in a fracture. J. Can. Pet. Technol. 49(02), 48–55 (2010). doi: 10.2118/133205-PA CrossRefGoogle Scholar
  31. Sisavath, S., Al-Yaarubi, A., Pain. C.C., Zimmerman, R.W.: A simple model for deviations from the cubic law for a fracture undergoing dilation or closure. In: Thermo-Hydro-Mechanical Coupling in Fractured Rock, pp. 1009–1022. Springer, Berlin (2003). doi: 10.1007/978-3-0348-8083-1_14
  32. Watanabe, N., Sakurai, K., Ishibashi, T., Ohsaki, Y., Tamagawa, T., Yagi, M., Tsuchiya, N.: New \(\nu \)-type relative permeability curves for two-phase flows through subsurface fractures. Water Resour. Res. 51(4), 2807–2824 (2015). doi: 10.1002/2014WR016515 CrossRefGoogle Scholar
  33. Ye, Z., Liu, H.H., Jiang, Q., Liu, Y., Cheng, A.: Two-phase flow properties in aperture-based fractures under normal deformation conditions: Analytical approach and numerical simulation. J. Hydrol. 545, 72–87 (2017). doi: 10.1016/j.jhydrol.2016.12.017 CrossRefGoogle Scholar
  34. Yu, C.: A simple statistical model for transmissivity characteristics curve for fluid flow through rough-walled fractures. Transp. Porous Med. 108(3), 649–657 (2015). doi: 10.1007/s11242-015-0493-x CrossRefGoogle Scholar
  35. Zimmerman, R.W., Kumar, S., Bodvarsson, G.: Lubrication theory analysis of the permeability of rough-walled fractures. Int. J. Rock Mech. Min. 28(4), 325–331 (1991). doi: 10.1016/0148-9062(91)90597-F CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of Chemical and Petroleum EngineeringSharif University of TechnologyTehranIran
  2. 2.Department of Chemical and Biomolecular EngineeringRice UniversityHoustonUSA
  3. 3.Sharif Upstream Petroleum Research Institute, Department of Chemical and Petroleum EngineeringSharif University of TechnologyTehranIran

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